Simple Proof Methods

Let's show the validity of another theorem using a proof by contradiction again.

Theorem. [Prime divisibility] A prime number isn't divisible by any other prime number.

Proof. Let $p \in \mathbb{P}$ be a prime number. To obtain a contradiction, assume that $p$ is divisible by some prime $q$ that isn't equal to $p$. Since $q$ is a prime, it means that $q > 1$ (because $1$ isn't a prime number, and prime numbers are always positive integers [other than $1$]).

This is a contradiction to the fact that $p$ is a prime, because, according to the definition of prime numbers on slide 44, the only numbers that divide $p$ are $1$ and $p$, so it's false that $q$, where $q \neq p$ and $q \neq 1$, divides $p$.

We, therefore, conclude that a prime number isn't divisible by any other prime number.◼